忘れないように、私は彼の電話番号を書き留めた。

、。

I wrote down his phone number so I wouldn't forget.

Literal

Forget-not [so-that-ように], I [topic-は] his phone-number [object-を] noted-down.

Same ~ないように + 書き留めた pattern. 私は makes the subject explicit. 電話番号 (phone number) is a standard compound. The comma after ように is a natural breath pause separating the purpose clause from the main clause.